More can be written, but for the overall question, you might like to start with the FAQ section "The Fall of the Western Roman Empire". There's another discussion off on another FAQ page under "Rome": 'How accurate is the popular view that "uncontrolled immigration led to the fall of the Roman Empire"?' by /u/Iguana_on_a_stick.

This is not to discourage discussion. More questions, data and debate are welcome.

To start, there was no one factor that directly caused the fall of the Western Roman Empire. It was a combination of several different things such as: the arrival of the Huns, the mass migrations of barbarians into the Empire, Rome's inability to remove these peoples from the Empire, etc. It should also be noted that Western Roman Empire fell via a military coup rather than being conquered by an outside force. Odoacer, who was an officer in the Roman military, overthrew Romulus Agustus, and took control of Italy (which was all that remained of the Western Empire by that point). He then refused the imperial standards that were sent to him by Emperor Zeno, and instead proclaimed himself as the King of Italy. You are right however when you say that the Empire's military strength had not decreased very much between the 4th and 5th centuries. It was still perfectly capable of defeating the migrating barbarians in battles, the problem was more so the matter of removing so many people from within the Empire. The large number of usurpers during the 5th century certainly played a role in this as well, and it likely made the reigning emperors more cautious when it came to deploying the army outside of Italy. Rome was not, however, lacking when it came to good leadership in this period. While there were some ineffectual emperors, such as Honorius, there were several talented generals and Emperors. Men like Emperor Majorian, and the generals Stilicho and Flavius Aetius (both of whom would be named Magister Militum during their respective careers). I personally study the role that diferent barbarian groups played in the downfall of the Empire, due to Rome's inability to remove them so that is what my answer will mostly be focused on.

While Theodosius' son Honorius, though this applies to both of them really, was a fairly ineffectual ruler, especially since he was only 11 when he became the Western emperor, he was lucky enough to have Stilicho who had risen to the rank of the magister militum during the reign of Theodosius. Stilicho was able to defeat Alaric and his Goths on multiple occasions when they attempted to invade Italy, though he was never able to score a truly decisive victory, and also defeated Radagaisus who invaded Italy with an extremely large force of Goths in 405. Stilicho also played a role in instigating the rebellion of Alaric and his Gothic troops. When Theodosius died he left Stilicho in charge of Honorius, Stilicho, however, claimed that he had also been left in charge of Arcadius. One of the main problems with this claim though, is that Arcadius was already 18 when his father died. On top of this Arcadius had little to no interest in actually being emperor and was easily controlled by one of his court officials, Rufinus, who then refuted Stilicho's claim. This also led to Stilicho to make multiple attempts to take the province of Illyricum by force. This was partially responsible for Alaric's rebellion.

Alaric took advantage of the political turmoil and took his Gothic troops from Italy to the Balkans where they proceded to ravage Greece. He may have been bribed by Rufinus to stop ravaging Rome, this is unclear, but by 397 he was most certainly serving the Eastern Empire, as he was made the magister militum of Illyricum. This may have been done as a way to use Alaric and his Goths to deter Stilicho from invading Illyricum again. If this was case then it did not work exactly as planned, as in 397 Stilicho once again attempted to conquer the province for the Western Empire. Stilicho and Alaric met in battle but it resulted in a standstill, and Stilicho had to withdraw his forces back to Italy. Between these events and 401 there was growing turmoil in Constantinople. A rebellion broke out among the Goths in Asia Minor in 399 and then they defeated the army that was sent to put them down. Gainas, who was also a Goth, was then named magister militum and was sent to defeat the rebels. He decided to join them howevr, possibly after having been defeated himself, and demanded that Arcadius exile the eunuch Eutropius and give him Eutropius' military office. Aracdius acquiesced, he most likely had little choice in the matter, but shortly afterwards the people of Constantinople rose up and slaughtered a large number of Gainas' Gothic troops. Arcadius then sent Fravitta, also a Goth, to defeat the rest of Gainas' men, and Gainas was defeated and killed. This caused several problems for Alaric. Firstly his position was no longer secure because he had gained it from Eutropius, and also because the the turmoil that had been caused by several prominent Goths. The Eastern Empire may have also returned the province that he was stationed in to the West, meaning that Alaric no longer had a military command in a province that was now controlled by his enemy Stilicho. In addition to this Fravitta was murdered soon after he defeated Gainas. If Alaric were to lose his title then he also would have lost his ability to feed and pay his men. These factors were likely the main driving force behind Alaric's rebellion and invasion of Italy in 401.